Difference between revisions of "Sequence"

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This naturally then generalises to [[Indexing set|indexing sets]]
 
This naturally then generalises to [[Indexing set|indexing sets]]
  
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==Notation==
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To specify that the points of a sequence, the {{M|x_i}} are from a space, {{M|X}} we may write:
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* {{M|1=(x_n)^\infty_{n=1}\subseteq X}}
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This is an abuse of notation, as {{M|1=(x_n)^\infty_{n=1} }} is not a subset of {{M|X}}. It plays on:
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* {{M|1=[(x_n)^\infty_{n=1}\subseteq X]\iff[x\in(x_n)_{n=1}^\infty\implies x\in X]}}
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Note that the elements of {{M|1=(x_n)_{n=1}^\infty}} are ether:
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* Elements of a [[Relation|relation]] (if we consider the sequence as a [[Function|mapping]]) or
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** So using this, {{M|1=x\in(x_n)_{n=1}^\infty}} may look like {{M|1=x=(a,b)}} (indicating {{M|1=f(a)=b}}) which is an [[Ordered pair]], not in {{M|X}}
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* Elements of a [[Tuple|tuple]] (which is a generalisation of [[Ordered pair|ordered pairs]] where (usually) {{M|1=(a,b)=\{\{a\},\{a,b\}\} }}
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** So using this, {{M|1=x\in(x_n)_{n=1}^\infty}} may indeed look like {{M|1=x=\{\{a\},\{a,b\}\}\notin X}}
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'''As such the notation {{M|1=(x_n)^\infty_{n=1}\subseteq X}} having no ''other'' sensible meaning''' is a notation to say that {{M|1=\forall i[x_i\in X]}}
 
==Subsequence==
 
==Subsequence==
 
Given a sequence {{M|1=(x_n)_{n=1}^\infty}} we define a ''subsequence of {{M|1=(x_n)^\infty_{n=1} }}''<ref name="Analysis"/> as a sequence:
 
Given a sequence {{M|1=(x_n)_{n=1}^\infty}} we define a ''subsequence of {{M|1=(x_n)^\infty_{n=1} }}''<ref name="Analysis"/> as a sequence:
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* [[Monotonic sequence]]
 
* [[Monotonic sequence]]
 
* [[Bolzano-Weierstrass theorem]]
 
* [[Bolzano-Weierstrass theorem]]
* [[Cauchy criterion for convergence]]
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* [[Cauchy sequence]] (Alternatively: [[Cauchy criterion for convergence]])
 
* [[Convergence of a sequence]]
 
* [[Convergence of a sequence]]
  

Revision as of 13:56, 9 July 2015

A sequence is one of the earliest and easiest definitions encountered, but I will restate it.

I was taught to denote the sequence {a1,a2,...} by {an}n=1 however I don't like this, as it looks like a set. I have seen the notation (an)n=1 and I must say I prefer it. This notation is inline with that of a tuple which is a generalisation of an ordered pair.

Definition

Formally a sequence (Ai)i=1 is a function[1][2], f:NS where S is some set. For a finite sequence it is simply f:{1,...,n}S. Now we can write:

  • f(i):=Ai

This naturally then generalises to indexing sets

Notation

To specify that the points of a sequence, the xi are from a space, X we may write:

  • (xn)n=1X

This is an abuse of notation, as (xn)n=1 is not a subset of X. It plays on:

  • [(xn)n=1X][x(xn)n=1xX]

Note that the elements of (xn)n=1 are ether:

  • Elements of a relation (if we consider the sequence as a mapping) or
    • So using this, x(xn)n=1 may look like x=(a,b) (indicating f(a)=b) which is an Ordered pair, not in X
  • Elements of a tuple (which is a generalisation of ordered pairs where (usually) (a,b)={{a},{a,b}}
    • So using this, x(xn)n=1 may indeed look like x={{a},{a,b}}X

As such the notation (xn)n=1X having no other sensible meaning is a notation to say that i[xiX]

Subsequence

Given a sequence (xn)n=1 we define a subsequence of (xn)n=1[2] as a sequence:

  • k:NN which operates on an nN with nkn:=k(n) where:
    • kn is increasing, that means knkn+1

We denote this:

  • (xkn)n=1

See also

References

  1. Jump up p46 - Introduction To Set Theory, third edition, Jech and Hrbacek
  2. Jump up to: 2.0 2.1 p11 - Analysis - Part 1: Elements - Krzysztof Maurin